Question

If a,b,c are distinct, show that `[[1,1,1],[a,b,c],[a^3,b^3,c^3]] = (b-c) * (c-a)*(a-b)(a+b+c)`

Answer 1

`L.H.S. = |[1,1,1],[a,b,c],[a^3,b^3,c^3]|`
Applying `C_1->C_1- C_2 and C_2->C_2-C_3`
`=|[0,0,1],[a-b,b-c,c],[a^3-b^3,b^3-c^3,c^3]|`
`=|[0,0,1],[a-b,b-c,c],[(a-b)(a^2+ab+b^2),(b-c)(b^2+bc+c^2),c^3]|`
`=(a-b)(b-c)|[0,0,1],[1,1,c],[a^2+ab+b^2,b^2+bc+c^2,c^3]|`
`=(a-b)(b-c)[b^2+bc+c^2 - a^2- ab-b^2]`
`=(a-b)(b-c)[bc - ab +c^2 - a^2]`
`=(a-b)(b-c)[b(c-a)+(c+a)(c-a)]`
`=(a-b)(b-c)(c-a)(a+b+c) = R.H.S.`